An Algorithm for a Positive Solution of Arbitrary Fully Fuzzy Linear System

Malkaw, G.; Ahmad, Nazihah; Ibrahim, Haslinda

doi:10.1007/s10598-015-9283-0

Cited by 12 publications

(5 citation statements)

References 27 publications

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“…This is because a near-zero fuzzy number cannot be defined in the form of (m, α, β), unlike a positive or negative fuzzy number could. Therefore, a new form of multiplication arithmetic operator has been introduced by [24] which adapted the system of min-max function.…”

Section: Triangular Fuzzy Number Definitionmentioning

confidence: 99%

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A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Daud

Ahmad

Malkawi³

2021

TELKOMNIKA

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Matrix equations have its own important in the field of control system engineering particularly in the stability analysis of linear control systems and the reduction of nonlinear control system models. There are certain conditions where the classical matrix equation are not well equipped to handle the uncertainty problems such as during the process of stability analysis and reduction in control system engineering. In this study, an algorithm is developed for solving fully fuzzy matrix equation particularly forÃXB −X =C, where the coefficients of the equation are in near-zero fuzzy numbers. By modifying the existing fuzzy multiplication arithmetic operators, the proposed algorithm exceeds the positive restriction to allow the near-zero fuzzy numbers as the coefficients. Besides that, a new fuzzy subtraction arithmetic operator has also been proposed as the existing operator failed to satisfy the both sides of the near-zero fully fuzzy matrix equation. Subsequently, Kronecker product and V ec-operator are adapted with the modified fuzzy arithmetic operator in order to transform the fully fuzzy matrix equation to a fully fuzzy linear system. On top of that, a new associated linear system is developed to obtain the final solution. A numerical example and the verification of the solution are presented to demonstrate the proposed algorithm.This is an open access article under the CC BY-SA license.

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Section: Triangular Fuzzy Number Definitionmentioning

confidence: 99%

“…Theorem 1. [24] Consider an arbitrary fuzzy numberM = (m, α, β) and a positive fuzzy number N = (n, γ, δ), i. IfM is positive, then the following inequalities are satisfied:…”

Section: Definitionmentioning

confidence: 99%

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Daud

Ahmad

Malkawi³

2021

TELKOMNIKA

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“…Theorem 1 [24] Consider an arbitrary fuzzy numberM = (m a , α a , β a ) and positive fuzzy solutionX = (m x , α x , β x ) .…”

Section: Preliminariesmentioning

confidence: 99%

A new solution of pair matrix equations with arbitrary triangular fuzzy numbers

Daud¹,

Ahmad²,

Malkawi³

2019

Turk J Math

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Pair matrix equations have numerous applications in control system engineering, such as for stability analysis of linear control systems and also for reduction of nonlinear control system models. There are some situations in which the classical pair matrix equations are not well equipped to deal with the uncertainty problem during the process of stability analysis and reduction in control system engineering. Thus, this study presents a new algorithm for solving fully fuzzy pair matrix equations where the parameters of the equations are arbitrary triangular fuzzy numbers. The fuzzy Kronecker product and fuzzy V ec-operator are employed to transform the fully fuzzy pair matrix equations to a fully fuzzy pair linear system. Then a new associated linear system is developed to convert the fully fuzzy pair linear system to a crisp linear system. Finally, the solution is obtained by using a pseudoinverse method. Some related theoretical developments and examples are constructed to illustrate the proposed algorithm. The developed algorithm is also able to solve the fuzzy pair matrix equation.

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“…There are many authors have written about developing fuzzy analysis, significantly solving fully fuzzy linear and nonlinear systems. Different methods have been used to solve these two types of fully fuzzy systems, including a method for computing the positive solution of a fully fuzzy linear system (Ezzati et al, 2012); a method used to solve a fully fuzzy linear system via decomposing the symmetric coefficient matrix into two equations systems with the Cholesky method (Senthilkumar and Rajendran, 2011); the Jacobi iteration method for solving a fully fuzzy linear system with fuzzy arithmetic (Marzuki, 2015) and triangular fuzzy number (Megarani et al, 2022); a method for finding a positive solution for an arbitrary fully fuzzy linear system with a one-block matrix (Malkawi et al, 2015a); the singular value decomposition method for solving a fully fuzzy linear system (Marzuki et al, 2018); the Gauss-Seidel method for solving a fully fuzzy linear system via alternative multi-playing triangular fuzzy numbers (Deswita and Mashadi, 2019); the Jacobi, Gauss-Seidel, and SOR iterative methods for solving linear fuzzy systems (Inearat and Qatanani, 2018);a linear programming approach utilizing equality constraints to find non-negative fuzzy numbers (Otadi and Mosleh, 2012); combining interval arithmetic with trapezoidal fuzzy numbers to solve a fully fuzzy linear system (Siahlooei and Fazeli, 2018); using an ST decomposition with trapezoidal fuzzy numbers to solve dual fully fuzzy linear systems via alternative fuzzy algebra (Safitri and Mashadi, 2019), using LU factorizations of coefficient matrices for trapezoidal fuzzy numbers to solve dual fully fuzzy linear systems (Marni et al, 2018); combining QR decomposition with trapezoidal fuzzy numbers (Gemawati et al, 2018); and using an ST decomposition with trapezoidal fuzzy numbers to solve a dual fully fuzzy linear system (Jafarian, 2016). In the case of fully fuzzy linear matrix equations, several studies have identified methods for solving them, including a method that utilizes fully fuzzy Sylvester matrix equations (Daud et al, 2018;Elsayed et al, 2022;He et al, 2018;Malkawi et al, 2015b), and a method that finds fuzzy approximate solutions (Guo and Shang, 2013).…”

Section: Introductionmentioning

confidence: 99%

Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method

Zakaria

Megarani

Faisol

et al. 2023

Int. j. math. eng. manag. sci.

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Fuzzy numbers have many applications in various mathematical models in both linear and nonlinear forms. In the form of a nonlinear system, fuzzy nonlinear equations can be constructed in the form of matrix equations. Unfortunately, the matrix equations used to solve the problem of dual fully fuzzy nonlinear systems are still relatively few found in the publication of research results. This article attempts to solve a dual fully fuzzy nonlinear equation system involving triangular fuzzy numbers using Broyden’s method. This article provides the pseudocode algorithm and the implementation of the algorithm into the MATLAB program for the iteration process to be carried out quickly and easily. The performance of the given algorithm is the fastest in finding system solutions and provides a minimum error value.

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An Algorithm for a Positive Solution of Arbitrary Fully Fuzzy Linear System

Cited by 12 publications

References 27 publications

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

A new solution of pair matrix equations with arbitrary triangular fuzzy numbers

Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method

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